Equations
(these are the Gibbs equations)
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T dS = dU + P dV
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T dS = dH − V dP
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Details
The first of the relations above can be derived by considering a simple compressible substance in the absence of motion or gravitational effects. The first law for a change of state under these conditions can be written:
δQ = dU + δW
It is of interest to first deal with the changes of state in which the state of the substance can be identified at all times. Thus, a quasi-equilibrium process must be considered. For a reversible process of a simple compressible substance:
Substituting these relations into the first-law equation results in:
(Eq1) T dS = dU + P dV
Note that this equation was derived by assuming a reversible process. The equation can therefore be integrated for any reversible process, for during such a process the state of the substance can be identified at any point during the process. Eq1 deals only with properties. Suppose for an irreversible process taking place between given initial and final states. The properties of a substance depend only on the state, and therefore the changes in the properties during a given change of state are the same for an irreversible process as for a reversible process. Therefore, the Eq1 is often applied to an irreversible process between two given states, but the integration of the equation is performed along a reversible path between the same two states. Since the enthalpy is defined as:
H = U + PV
it follows that:
dH = dU + P dV + V dP
substituting this relation into Eq1:
(Eq2) T dS = dH – V dP
Eq1 and Eq2 are two forms of the thermodynamic property relation and are frequently called Gibbs equations. These equations can also be written for a unit mass:
T ds = du + P dv
and
T ds = dh – v dP
If substances of fixed composition are considered other than a simple compressible substance, "T dS" equations can be written other than those just given for a simple compressible substance. For a reversible process the following equation may apply for work:
δW = P dV – Ʈ dL – ℒ dA – ℰ dZ + ⋅⋅⋅
It follows that a more general expression for the thermodynamic property relation would be:
T ds = dU + P dV – Ʈ dL – ℒ dA – ℰ dZ + ⋅⋅⋅
Note: (big letter) = (mass)*(small letter), ex: W = mw, this is applicable for work (W, w), heat (Q, q), enthalpy (H, h), entropy (S, s). The pattern here is the energy terms with the exception of volume/specific volume (V, v)